ON THE CHEBYSHEV POLYNOMIAL COEFFICIENT PROBLEM OF BI-BAZILEVI C FUNCTIONS

Abstract

A function said to be bi-Bazilevic in the open unit disk U if both the function and its inverse are Bazilevic there. In this paper, we will study a newly constructed class of bi-Bazilevic functions. Furthermore, we establish Chebyshev polynomial bounds for the coecients, and get Fekete-Szego inequality, for the class B ; t .

Keywords

• Chebyshev polynomials,analytic and univalent functions,bi-univalent functions,bi-Bazilevic functions,coecient bounds,subordination,Fekete-Szego inequality.

References

1. Altınkaya, S¸. and Yal¸cın, S., (2014), Fekete-Szeg¨o inequalities for certain classes of bi-univalent func- tions, Internat. Scholar. Res. Notices, (2014) Article ID 327962, pp. 1-6.
2. Altınkaya, S¸. and Yal¸cın, S., (2015), Coefficient estimates for two new subclasses of bi-univalent functions with respect to symmetric points, J. Funct. Spaces, Article ID 145242 pp. 1-5.
3. Altınkaya, S¸. and Yal¸cın, S., (2015), Faber polynomial coefficient bounds for a subclass of bi-univalent functions, C.R. Acad. Sci. Paris, Ser. I, 353, pp. 1075-1080.
4. Altınkaya, S¸. and Yal¸cın, S., (2015), Coefficient bounds for a subclass of bi-univalent functions, TWMS J. Pure Appl. Math., 6, pp. 180-185.
5. Altınkaya, S¸. and Yal¸cın, S., (2016), On the Chebyshev polynomial bounds for classes of univalent functions, Khayyam J. Math., 2, pp. 1-5.
6. Brannan, D. A. and Taha, T. S., (1986), On some classes of bi-univalent functions, Stud. Univ. Babe¸s-Bolyai Math., 31, pp. 70-77.
7. Brannan, D. A. and Clunie, J. G., (1980), Aspects of comtemporary complex analysis, (Proceedings of the NATO Advanced Study Instute Held at University of Durham:July 1-20, 1979), New York: Academic Press.
8. C¸ a˘glar, M., Deniz, E. and Srivastava, H.M., (2017), Second Hankel determinant for certain subclasses of bi-univalent functions, Turk J Math, 41, pp. 694-706.

9. Dziok, J., Raina, R. K. and Sokol, J., (2015), Application of Chebyshev polynomials to classes of analytic functions, C. R. Acad. Sci. Paris, Ser. I, 353, pp. 433-438.
10. Duren, P. L., (1983), Univalent Functions, Grundlehren der Mathematischen Wissenschaften, Springer, New York, USA, 259.
11. Doha, E. H., (1994), The first and second kind Chebyshev coefficients of the moments of the general- order derivative of an infinitely differentiable function, Intern. J. Comput. Math., 51, pp. 21-35.
12. Fekete, M. and Szeg¨o, G., (1933), Eine Bemerkung ber Ungerade Schlichte Funktionen, J. London Math. Soc., [s1-8 (2)], pp. 85-89.
13. Hamidi, S. G. and Jahangiri, J. M., (2014), Faber polynomial coefficient estimates for analytic bi- close-to-convex functions, C. R. Acad. Sci. Paris, Ser. I, 352, pp. 17-20.
14. Hayami, T. and Owa, S., (2012), Coefficient bounds for bi-univalent functions, Pan Amer. Math., 22, pp. 15-26.
15. Kowalczyk, B., Lecko, A. and Srivastava, H. M., (2017), A note on the Fekete-Szeg¨o problem for close-toconvex functions with respect to convex functions, Publications de l’Institut Math´ematique, Nouvelle S´erie, 101[115], pp. 143-149.
16. Kim, Y.C. and Srivastava, H. M., (2006), The Hardy space for a certain subclass of Bazileviˇc functions, Appl. Math. Comput., 183, pp. 1201-1207
17. Lewin, M., (1967), On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 18, pp. 63-68.
18. Mason, J. C., (1967), Chebyshev polynomials approximations for the L-membrane eigenvalue problem, SIAM J. Appl. Math., 15, pp. 172-186.
19. Netanyahu, E., (1969), The minimal distance of the image boundary from the origin and the second coefficient of a univalent function in |z| < 1, Arch. Ration. Mech. Anal., 32, pp. 100-112.
20. Singh, R., (1973), On Bazilevi˘c Functions, Proc. Amer. Math. Soc., 38, pp. 261-271.
21. Srivastava, H. M., Mishra, A. K. and Gochhayat, P., (2010), Certain subclasses of analytic and bi- univalent functions, Appl. Math. Lett., 23, pp. 1188-1192.
22. Srivastava, H. M. and Bansal, D., (2015), Coefficient estimates for a subclass of analytic and bi- univalent functions, J. Egyptian Math. Soc., 23, pp. 242-246.
23. Srivastava, H. M., Bulut, S., C¸ a˘glar, M. and Ya˘gmur, N., (2013), Coefficient estimates for a general subclass of analytic and bi-univalent functions, Filomat, 27, pp. 831-842.
24. Srivastava, H. M., Joshi, S. B., Joshi, S. S. and Pawar, H., (2016), Coefficient estimates for certain subclasses of meromorphically bi-univalent functions, Palest. J. Math., 5, pp. 250-258.
25. Srivastava, H. M., Gaboury, S. and Ghanim, F., (2016), Initial coefficient estimates for some subclasses of m-fold symmetric bi-univalent functions, Acta Math. Sci. Ser. B Engl. Ed., 36, pp. 863–871.
26. Srivastava, H. M., Gaboury, S. and Ghanim, F., (2017), Coefficient estimates for some general sub- classes of analytic and bi-univalent functions, Afrika Matematika, 28, pp. 693-706.
27. Srivastava, H. M., Mishra, A. K. and Das, M. K., (2001), The Fekete-Szeg¨o problem for a subclass of close-to-convex functions, Complex Var. Theory Appl., 44, pp. 145-163.
28. Srivastava, H. M., Murugusundaramoorthy, G. and Vijaya, K., (2013), Coefficient estimates for some families of bi-Bazileviˇc functions of the Ma-Minda type involving the Hohlov operator, J. Class. Anal., 2, pp. 167-181.
29. Srivastava, H. M., S¨umer Eker, S. and Ali, R. M., (2015), Coefficient bounds for a certain class of analytic and bi-univalent functions, Filomat, 29, pp. 1839-1845.
30. Srivastava, H. M., Sivasubramanian, S. and Sivakumar, R., (2014), Initial coefficient bounds for a subclass of m-fold symmetric bi-univalent functions, Tbilisi Math. J., 7, pp. 1-10.
31. Tang, H., Srivastava, H. M., Sivasubramanian, S. and Gurusamy, P., (2016), The Fekete-Szeg¨o func- tional problems for some subclasses of m-fold symmetric bi-univalent functions, Journal of Mathe- matical Inequalities, 10, pp. 1063-1092.
32. Xu, Q.-H., Gui, Y.-C. and Srivastava, H. M., (2012), Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Appl. Math. Lett., 25, pp. 990-994.
33. Xu, Q.-H., Xiao, H.-G. and Srivastava, H. M., (2012), A certain general subclass of analytic and bi-univalent functions and associated coefficient estimates problems, Appl. Math. Comput., 218, pp. 11461-11465.
34. Zaprawa, P., (2014), On Fekete-Szeg problem for classes of bi-univalent functions, Bull. Belg. Math. Soc. Simon Stevin, 21, pp. 169-178.
35. Zireh, A., Hajiparvaneh, S. and Bulut, S., (2016), Faber polynomial coefficient estimates for a com- prehensive subclass of analytic bi-univalent functions defined by subordination, Bull. Belg. Math. Soc. Simon Stevin, 23, pp. 487-504.
36. S¸ahsene Altınkaya for the photography and short autobiography, see TWMS J. App. Eng. Math.V.8, N.1a, 2018.